Standard Ensemble Kalman Filter Research Articles

A Quantile-Conserving Ensemble Filter Framework. Part II: Regression of Observation Increments in a Probit and Probability Integral Transformed Space

Abstract Traditional ensemble Kalman filter data assimilation methods make implicit assumptions of Gaussianity and linearity that are strongly violated by many important Earth system applications. For instance, bounded quantities like the amount of a tracer and sea ice fractional coverage cannot be accurately represented by a Gaussian that is unbounded by definition. Nonlinear relations between observations and model state variables abound. Examples include the relation between a remotely sensed radiance and the column of atmospheric temperatures, or the relation between cloud amount and water vapor quantity. Part I of this paper described a very general data assimilation framework for computing observation increments for non-Gaussian prior distributions and likelihoods. These methods can respect bounds and other non-Gaussian aspects of observed variables. However, these benefits can be lost when observation increments are used to update state variables using the linear regression that is part of standard ensemble Kalman filter algorithms. Here, regression of observation increments is performed in a space where variables are transformed by the probit and probability integral transforms, a specific type of Gaussian anamorphosis. This method can enforce appropriate bounds for all quantities and deal much more effectively with nonlinear relations between observations and state variables. Important enhancements like localization and inflation can be performed in the transformed space. Results are provided for idealized bivariate distributions and for cycling assimilation in a low-order dynamical system. Implications for improved data assimilation across Earth system applications are discussed.

A reduced basis ensemble Kalman method

In the process of reproducing the state dynamics of parameter dependent distributed systems, data from physical measurements can be incorporated into the mathematical model to reduce the parameter uncertainty and, consequently, improve the state prediction. Such a data assimilation process must deal with the data and model misfit arising from experimental noise as well as model inaccuracies and uncertainties. In this work, we focus on the ensemble Kalman method (EnKM), a particle-based iterative regularization method designed for a posteriori analysis of time series. The method is gradient free and, like the ensemble Kalman filter (EnKF), relies on a sample of parameters or particle ensemble to identify the state that better reproduces the physical observations, while preserving the physics of the system as described by the best knowledge model. We consider systems described by parameterized parabolic partial differential equations and employ model order reduction techniques to generate surrogate models of different accuracy with uncertain parameters. Their use in combination with the EnKM involves the introduction of the model bias which constitutes a new source of systematic error. To mitigate its impact, an algorithm adjustment is proposed accounting for a prior estimation of the bias in the data. The resulting RB-EnKM is tested in different conditions, including different ensemble sizes and increasing levels of experimental noise. The results are compared to those obtained with the standard EnKF and with the unadjusted algorithm.

Fast and Accurate Estimation of Evapotranspiration for Smart Agriculture

AbstractThe ability to quantify evapotranspiration (ET) is crucial for smart agriculture and sustainable groundwater management. Efficient ET estimation strategies often rely on the vertical‐flow assumption to assimilate data from soil‐moisture sensors. While adequate in some large‐scale applications, this assumption fails when the horizontal component of the local flow velocity is not negligible due to, for example, soil heterogeneity or drip irrigation. We present novel implementations of the ensemble Kalman filter (EnKF) and the maximum likelihood estimation (MLE), which enable us to infer spatially varying ET rates and root water uptake profiles from soil‐moisture measurements. While the standard versions of EnKF and MLE update the predicted soil moisture prior to computing ET, ours treat the ET sink term in Richards' equation as an updatable observable. We test the prediction accuracy and computational efficiency of our methods in a setting representative of drip irrigation. Our strategies accurately estimate the total ET rates and root‐uptake profiles and do so up to two‐orders of magnitude faster than the standard EnKF.

The Improved Winter Wheat Yield Estimation by Assimilating GLASS LAI Into a Crop Growth Model With the Proposed Bayesian Posterior-Based Ensemble Kalman Filter

Data assimilation has been demonstrated as the potential crop yield estimation approach. Accurate quantification of model and observation errors is the key to determining the success of a data assimilation system. However, the crop growth model error is not fully taken into account in most of the previous studies. The objective of this study is to better quantify the model uncertainty in the data assimilation system. Firstly, we calibrated a crop growth model and inferred its posterior uncertainty based on the Global LAnd Surface Satellite (GLASS) 250-m LAI product, regional statistical data, station observations, and field measurements with a Markov chain Monte Carlo (MCMC) method. Secondly, the model posterior uncertainty was used in the Ensemble Kalman Filter (EnKF) algorithm to better characterize the ensemble distribution of model errors. Our results indicated the proposed Bayesian posterior-based EnKF can improve the accuracy of winter wheat yield estimation at both the point scale (the coefficient of determination R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> value increasing from 0.06 to 0.41, the mean absolute percentage error MAPE value decreasing from 12.65% to 7.82%, and the root mean square error RMSE value decreasing from 987 to 688 kg∙ha <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ) and the regional scale (R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> value from 0.30 to 0.57, MAPE value from 19.67% to 10.13%, and RMSE value from 1275 to 695 kg∙ha <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ) compared with the open-loop estimation. Our analysis also indicated that the Bayesian posterior-based EnKF can perform better compared to the standard Gaussian perturbation-based EnKF. The proposed framework provides an important reference for crop yield estimation at the regional scale in similar agricultural landscapes worldwide.

Particle network EnKF for large-scale data assimilation

The Ensemble Kalman filter (EnKF) is a classic method of data assimilation. For distributed sampling, the conventional EnKF usually requires a centralized server to integrate the predictions of all particles or a fully-connected communication network, causing traffic jams and low bandwidth utilization in high-performance computing. In this paper, we propose a novel distributed scheme of EnKF based on network setting of sampling, called Particle Network EnKF. Without a central server, every sampling particle communicates with its neighbors over a sparsely connected network. Unlike the existing work, this method focuses on the distribution of sampling particles instead of sensors and has been proved effective and robust on numerous tasks. The numerical experiments on the Lorenz-63 and Lorenz-96 systems indicate that, with proper communication rounds, even on a sparse particle network, this method achieves a comparable performance to the standard EnKF. A detailed analysis of effects of the network topology and communication rounds is performed. Another experiment demonstrating a trade-off between the particle homogeneity and performance is also provided. The experiments on the whole-brain neuronal network model show promises for applications in large-scale assimilation problems.

Improving surface soil moisture retrievals through a novel assimilation algorithm to estimate both model and observation errors

Soil moisture controls the land surface water and energy budget and plays a crucial role in land surface processes. Based on certain mathematical rules, data assimilation can merge satellite observations and land surface models, and produce spatiotemporally continuous profile soil moisture. The two mainstream assimilation algorithms (variational-based and sequential-based) both need model error and observation error estimates, which greatly impact the assimilation results. Moreover, the performance of land data assimilation relies heavily on the specification of model parameters. However, it is always challenging to specify these errors and model parameters. In this study, a dual-cycle assimilation algorithm was proposed for addressing the above issue. In the inner cycle, the Ensemble Kalman Filter (EnKF) is run with parameters of both model and observation operators and their errors, which are provided by the outer cycle. Both the analyzed state variable and the innovation are reserved at each analysis moment. In the outer cycle, the innovation time series kept by the inner cycle are fed into a likelihood function to adjust the values of parameters of both the model and observation operators and their errors through an optimization algorithm. A series of assimilation experiments were first performed based on the Lorenz-63 model. The results illustrate that the performance of the dual-cycle algorithm substantially surpasses those of both the classical parameter calibration and the standard EnKF. Subsequently, the Advanced Microwave Scanning Radiometer of earth Observing System (AMSR-E) brightness temperatures were assimilated into the simple biosphere model scheme version 2 (SiB2) with a radiative transfer model as the observation operator in two experimental areas, namely Naqu on the Tibetan Plateau and a Coordinate Enhanced Observing (CEOP) reference site in Mongolia. The results indicate that the dual-cycle assimilation algorithm can simultaneously estimate model parameters, observation operator parameters, model error, and observation error, thus improving surface soil moisture estimation in comparison with other assimilation algorithms. Since the dual-cycle assimilation algorithm can estimate the observation errors, it provides the potential for assimilating multi-source remote sensing data to generate physically consistent land surface state and flux estimates.

Estimation of Evapotranspiration Rates and Root Water Uptake Profiles From Soil Moisture Sensor Array Data

AbstractEvapotranspiration is arguably the least quantified component of the hydrologic cycle. We propose two complementary strategies for estimation of evapotranspiration rates and root water uptake profiles from soil‐moisture sensor‐array data. One is our implementation of ensemble Kalman filter (EnKF); it treats the evapotranspiration sink term in the Richards equation, rather than soil moisture, as the observable to update. The other is a maximum likelihood estimator (MLE) applied to the same observable; it is supplemented with the Fisher information matrix to quantify uncertainty in its predictions. We use numerical experiments to demonstrate the accuracy and computational efficiency of these techniques. We found our EnKF implementation to be two orders of magnitude faster than either the standard EnKF or MLE, and our MLE procedure to require an order of magnitude fewer iterations to converge than its counterpart applied to soil moisture. These findings render our methodologies a viable and practical tool for estimation of the root water uptake profiles and evaporation rates, with the MLE technique to be used when the prior knowledge about evapotranspiration at the site is elusive.

An efficient continuous data assimilation algorithm for the Sabra shell model of turbulence.

Complex nonlinear turbulent dynamical systems are ubiquitous in many areas of research. Recovering unobserved state variables is an important topic for the data assimilation of turbulent systems. In this article, an efficient continuous-in-time data assimilation scheme is developed, which exploits closed analytic formulas for updating the unobserved state variables. Therefore, it is computationally efficient and accurate. The new data assimilation scheme is combined with a simple reduced order modeling technique that involves a cheap closure approximation and noise inflation. In such a way, many complicated turbulent dynamical systems can satisfy the requirements of the mathematical structures for the proposed efficient data assimilation scheme. The new data assimilation scheme is then applied to the Sabra shell model, which is a conceptual model for nonlinear turbulence. The goal is to recover the unobserved shell velocities across different spatial scales. It is shown that the new data assimilation scheme is skillful in capturing the nonlinear features of turbulence including the intermittency and extreme events in both the chaotic and the turbulent dynamical regimes. It is also shown that the new data assimilation scheme is more accurate and computationally cheaper than the standard ensemble Kalman filter and nudging data assimilation schemes for assimilating the Sabra shell model with partial observations.

Integrating deep learning-based data assimilation and hydrogeophysical data for improved monitoring of DNAPL source zones during remediation

Data assimilation techniques allow the integration of multi-source data with physical modeling of dense nonaqueous phase liquid (DNAPL) source zones to monitor their evolution in time and thereby improve remediation efforts. The DNAPL source zone is governed by multiphase physics and its architecture is usually highly irregular and nonstationary. Traditional data assimilation methods use stationary models to describe these nonstationary DNAPL saturation fields, leading to significant errors in the estimation of unknown source zone architectures (SZAs). In this study, we propose a geostatistical data assimilation method to improve performance monitoring of DNAPL remediation. We estimate the DNAPL SZA after parameterizing the DNAPL saturation field using a deep learning method instead of a stationary statistical model. First, we train a convolutional variational autoencoder (CVAE) network to capture the physics of DNAPL infiltration and depletion. Then, we incorporate the CVAE network within the Ensemble Kalman filter (EnKF) to resolve the evolving DNAPL SZA by assimilating time-lapse hydrogeophysical measurements (i.e., oscillatory hydraulic tomography, downgradient dissolved DNAPL concentration, and electrical resistivity tomography). To assess the proposed CVAE-EnKF framework's performance, we conducted numerical experiments with evolving DNAPL SZAs in a 2D heterogeneous aquifer. Results show that the proposed framework significantly improved DNAPL saturation estimates throughout the source zone over time, compared to the standard EnKF method. CVAE-EnKF reduced the estimation error of DNAPL mass remediated by 51% from EnKF, along with better predictions of source zone longevity. It also better captured the morphology and center/spread of mass of the evolving remediated portions of the source zone and provided better estimates of pooled-DNAPL mass. Furthermore, the CVAE-EnKF method is more stable and converges faster than EnKF alone. Overall, the proposed data assimilation framework can improve real-time monitoring of DNAPL remediation and risk analysis.

Balanced data assimilation for highly oscillatory mechanical systems

Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, the ensemble Kalman filter also has limitations due to its inherent Gaussian and linearity assumptions. These limitations can manifest themselves in dynamically inconsistent state estimates. We investigate this issue in this paper for highly oscillatory Hamiltonian systems with a dynamical behavior which satisfies certain balance relations. We first demonstrate that the standard ensemble Kalman filter can lead to estimates which do not satisfy those balance relations, ultimately leading to filter divergence. We also propose two remedies for this phenomenon in terms of blended time-stepping schemes and ensemble-based penalty methods. The effect of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for two model scenarios. First, we consider balanced motion for highly oscillatory Hamiltonian systems and, second, we investigate thermally embedded highly oscillatory Hamiltonian systems. The first scenario is relevant for applications from meteorology while the second scenario is relevant for applications of data assimilation to molecular dynamics.

Training a convolutional neural network to conserve mass in data assimilation

Abstract. In previous work, it was shown that the preservation of physical properties in the data assimilation framework can significantly reduce forecast errors. Proposed data assimilation methods, such as the quadratic programming ensemble (QPEns) that can impose such constraints on the calculation of the analysis, are computationally more expensive, severely limiting their application to high-dimensional prediction systems as found in Earth sciences. We, therefore, propose using a convolutional neural network (CNN) trained on the difference between the analysis produced by a standard ensemble Kalman filter (EnKF) and the QPEns to correct any violations of imposed constraints. In this paper, we focus on the conservation of mass and show that, in an idealised set-up, the hybrid of a CNN and the EnKF is capable of reducing analysis and background errors to the same level as the QPEns.

An Efficient Bi-Gaussian Ensemble Kalman Filter for Satellite Infrared Radiance Data Assimilation

AbstractThe introduction of infrared water vapor channel radiance ensemble data assimilation (DA) has improved numerical weather forecasting at operational centers. Further improvements might be possible through extending ensemble data assimilation methods to better assimilate infrared satellite radiances. Here, we will illustrate that ensemble statistics under clear-sky conditions are different from cloudy conditions. This difference suggests that extending the ensemble Kalman filter (EnKF) to handle bi-Gaussian prior distributions may yield better results than the standard EnKF. In this study, we propose a computationally efficient bi-Gaussian ensemble Kalman filter (BGEnKF) to handle bi-Gaussian prior distributions. As a proof-of-concept, we used the 40-variable Lorenz 1996 model as a proxy to examine the impacts of assimilating infrared radiances with the BGEnKF and EnKF. A nonlinear observation operator that constructs radiance-like bimodal ensemble statistics was used to generate and assimilate pseudoradiances. Inflation was required for both methods to effectively assimilate pseudoradiances. In both 800- and 20-member experiments, the BGEnKF generally outperformed the EnKF. The relative performance of the BGEnKF with respect to the EnKF improved when the observation spacing and time between DA cycles (cycling interval) are increased from small values. The relative performance then degraded when observation spacing and cycling interval become sufficiently large. The BGEnKF generated less noise than the EnKF, suggesting that the BGEnKF produces more balanced analysis states than the EnKF. This proof-of-concept study motivates future investigation into using the BGEnKF to assimilate infrared observations into high-order numerical weather models.

Saturation-based localization within EnKF approach for history matching in SAGD processes

The history matching method with the ensemble Kalman filter (EnKF) technique has been successfully applied in the steam-assisted gravity drainage (SAGD) processes. In the application of the EnKF method, usually, large ensemble size is required to eliminate spurious correlations, which also increases the computation cost. In this paper, a newly defined saturation-based localized EnKF approach is proposed and proved to be much more efficient than the standard EnKF method. This approach is able to use a small ensemble to obtain the desired history match and prediction results. Therefore, the computation cost can be reduced. Simulation results from a large ensemble (1000 ensembles) size are used to conduct cross-covariance analysis to determine the regions of the localization function. Based on the positive covariance regions, the localization regions are defined. Oil saturation is used as localization scale to define the saturation-based localization function. Good history match results and prediction results are obtained by adopting saturation-based localized EnKF approach. A small ensemble (10 ensembles) size is employed to test the efficiency of the newly defined localization function. Without localization, the ensemble variability collapses quickly when using small ensembles which will have no data assimilation ability. The saturation-based localized EnKF approach with small ensembles (10 ensembles) can avoid ensemble variability collapse and obtain desired history matching results.

A Marginal Adjustment Rank Histogram Filter for Non-Gaussian Ensemble Data Assimilation

AbstractAn extension to standard ensemble Kalman filter algorithms that can improve performance for non-Gaussian prior distributions, non-Gaussian likelihoods, and bounded state variables is described. The algorithm exploits the capability of the rank histogram filter (RHF) to represent arbitrary prior distributions for observed variables. The rank histogram algorithm can be applied directly to state variables to produce posterior marginal ensembles without the need for regression that is part of standard ensemble filters. These marginals are used to adjust the marginals obtained from a standard ensemble filter that uses regression to update state variables. The final posterior ensemble is obtained by doing an ordered replacement of the posterior marginal ensemble values from a standard ensemble filter with the values obtained from the rank histogram method applied directly to state variables; the algorithm is referred to as the marginal adjustment rank histogram filter (MARHF). Applications to idealized bivariate problems and low-order dynamical systems show that the MARHF can produce better results than standard ensemble methods for priors that are non-Gaussian. Like the original RHF, the MARHF can also make use of arbitrary non-Gaussian observation likelihoods. The MARHF also has advantages for problems with bounded state variables, for instance, the concentration of an atmospheric tracer. Bounds can be automatically respected in the posterior ensembles. With an efficient implementation of the MARHF, the additional cost has better scaling than the standard RHF.

Non-Gaussian Deterministic Assimilation of Radar-Derived Precipitation Accumulations

Abstract Data assimilation (DA) approaches currently used for operational numerical weather prediction (NWP) generally assume that errors in the background state are Gaussian. At the same time, approaches that make no assumptions regarding the background state probability distribution are gaining attention in research. Most such approaches, including the particle filter, are ensemble DA methods that produce an ensemble of analysis states consistent with the background and observation distributions. The present study instead proposes a non-Gaussian deterministic (NGD) DA method for producing a single deterministic analysis state. Consequently, the usual challenge of maintaining an ensemble with sufficient spread and diversity is avoided. The NGD approach uses background ensembles generated by a standard ensemble Kalman filter. A series of noncycled DA experiments is conducted to evaluate the NGD approach for assimilating precipitation derived from North American weather radars to initialize limited-area deterministic forecasts. The resulting forecasts are compared with those produced using either a local ensemble transform Kalman filter (LETKF) deterministic analysis or latent heat nudging (LHN). The experimental results indicate that, for forecast lead times beyond 1.5 h, the NGD approach improves precipitation forecasts relative to LHN. The NGD approach also leads to better temperature and zonal wind forecasts at lead times up to 12 h when compared to those obtained with either LHN or the LETKF. For precipitation, the NGD and LETKF approaches produce forecasts that are of comparable quality. Finally, simple strategies are demonstrated that combine the NGD approach for assimilating radar-derived precipitation accumulations with the ensemble–variational approach for assimilating all other observations.

Shape and distributed parameter estimation for history matching using a modified Ensemble Kalman filter and level sets

ABSTRACTIn this paper, we introduce a novel method for the automatic history matching of 3D reservoirs with two lithofacies. The method simultaneously estimates distributed geological reservoir properties and facies boundaries by combining the general Ensemble Kalman Filter (EnKF) strategy with the level set method for shape propagation in 3D. Each iteration of this novel Ensemble Kalman Filter-Level Set (EnKF-LS) scheme combines the update of the boundaries of two lithofacies, represented by associated level set functions, with an update of the corresponding parameter fields, namely porosity and permeability. A narrow band technique is employed for restricting the shape motion to a neighbourhood of the current interfaces. The proposed algorithm is tested on a synthetic 3D reservoir model which is a modification of the popular SPE10 model. We use an initial ensemble of porosity and permeability fields created by a multiple-point statistics-based algorithm for a channelized test case. We compared the proposed method with an implementation of the standard EnKF algorithm and showed that the proposed EnKF-LS scheme not only provides more adequate results for such a channelized structure, but also produces a better data fit than the standard EnKF approach.

Desensitized Ensemble Kalman Filtering for Induction Motor Estimation

The ensemble Kalman filter (EnKF) is a random sampling method based on Monte Carlo and Kalman filter for the extremely high-dimensional nonlinear system. However, the uncertain parameters may be have unexpected effects on the state estimate problem. This paper introduces the desensitized optimal control methodology into the EnKF, and derives the algorithm formula of the proposed desensitized EnKF (DEnKF). A new desensitized cost function is designed by combing the mean square error cost function with the penalty cost function, and the optimal gain is obtained by minimizing the new cost function. The sensitivity of the state errors is propagated by the ensemble error matrix. To demonstrate the effectiveness of the proposed DEnKF, two numerical simulations for a falling body model and induction motor model are designed. The simulation results show that the proposed DEnKF could weaken the sensitivity of state estimate errors to the uncertain parameters and improves the state estimation accuracy, comparing with the standard EnKF.

Data Assimilation in Density‐Dependent Subsurface Flows via Localized Iterative Ensemble Kalman Filter

AbstractParameter estimation in variable‐density groundwater flow systems is confronted with challenges of strong nonlinearity and heavy computational burden. Relying on a variant of the Henry problem, we evaluate the performance of a domain localization scheme of the iterative ensemble Kalman filter in the framework of data assimilation settings for variable‐density groundwater flows in a seawater intrusion scenario. The performance of the approach is compared against (a) the corresponding domain localization scheme of the ensemble Kalman filter in its standard formulation as well as (b) a covariance localization scheme of the latter. The equivalent freshwater head, hf, and salinity, Sa, are set as the target state variables. The randomly heterogeneous field of equivalent freshwater hydraulic conductivity, Kf, is considered as the system parameter field. Density‐independent and density‐driven flow settings are considered to evaluate the assimilation results using various methods and data. When only hf data are assimilated, all tested approaches perform generally well and a localization scheme embedded in the iterative ensemble Kalman filter appears to consistently outperform the domain localized version of the standard ensemble Kalman filter (EnKF) in a density‐driven scenario; Dirichlet boundary conditions tend to show a more pronounced negative effect on estimating Kf for density‐independent than for density‐dependent flow conditions; hf data are more informative in a density‐dependent than in a density‐independent setting. The sole use of Sa information does not yield satisfactory updates of hf for the covariance localization scheme of the standard EnKF, while the sole use of hf does. The domain localization scheme leads to difficulties in the attainment of global filter convergence when only Sa data are used. A covariance localization scheme associated with a standard EnKF can significantly alleviate this issue.

A two-stage ensemble Kalman filter based on multiscale model reduction for inverse problems in time fractional diffusion-wave equations

Ensemble Kalman filter (EnKF) has been widely used in state estimation and parameter estimation for the dynamic system where observational data is obtained sequentially in time. Very burdened simulations for the forward problem are needed to update a large number of EnKF ensemble samples. This will slow down the analysis efficiency of the EnKF for large-scale and high dimensional models. To reduce uncertainty and accelerate posterior inference, a two-stage ensemble Kalman filter is presented to improve the sequential analysis of EnKF. It is known that the final posterior ensemble may be concentrated in a small portion of the entire support of the initial prior ensemble. It will be much more efficient if we first build a new prior by some partial observations, and construct a surrogate only over the significant region of the new prior. To this end, we construct a very coarse model using generalized multiscale finite element method (GMsFEM) and generate a new prior ensemble in the first stage. GMsFEM provides a set of hierarchical multiscale basis functions supported in coarse blocks. This gives flexibility and adaptivity to choosing degree of freedoms to construct a reduce model. In the second stage, we build an initial surrogate model based on the new prior by using GMsFEM and sparse generalized polynomial chaos (gPC)-based stochastic collocation methods. To improve the initial surrogate model, we dynamically update the surrogate model, which is adapted to the sequential availability of data and the updated analysis. The two-stage EnKF can achieve a better estimation than standard EnKF, and significantly improve the efficiency to update the ensemble analysis (posterior exploration). To enhance the applicability and flexibility in Bayesian inverse problems, we extend the two-stage EnKF to non-Gaussian models and hierarchical models. In the paper, we focus on the time fractional diffusion-wave models in porous media and investigate their Bayesian inverse problems using the proposed two-stage EnKF. A few numerical examples are carried out to demonstrate the performance of the two-stage EnKF method by taking account of parameter and structure inversion in permeability fields and source functions.

A stochastic well-test analysis on transient pressure data using iterative ensemble Kalman filter

Accurate estimation of the reservoir parameters is crucial to predict the future reservoir behavior. Well testing is a dynamic method used to estimate the petro-physical reservoir parameters through imposing a rate disturbance at the wellhead and recording the pressure data in the wellbore. However, an accurate estimation of the reservoir parameters from well-test data is vulnerable to the noise at the recorded data, the non-uniqueness of the obtained match, and the accuracy of the optimization algorithm. Different stochastic optimization methods have been applied to this address problem in the literature. In this study, we apply the recently developed iterative ensemble Kalman filter in the context of well-test analysis to infer reservoir parameters from the noisy recorded data. Since the introduction of the ensemble Kalman filter (EnKF) by Evensen in 1994 as a novel method for data assimilation, it has had enormous impact in many application domains because of its robustness and ease of implementation, and numerical evidence of its accuracy. While the objective of the standard EnKF approaches is to approximate the statistical properties of geological parameters conditioned to observation, via an ensemble, the objective of the iterative ensemble Kalman methods is to approximate the solution of inverse problems using a deterministic derivative-free iterative scheme. We conducted three case studies of the application of the iterative ensemble Kalman methods for a well-test analysis of a homogenous reservoir model, a dual-porosity heterogeneous system, and a faulted discontinuous reservoir. We demonstrated that the convergence occurs very rapidly almost at the first iterations contrary to the well-known particle swarm optimization algorithm. The maximum relative error for the simulated cases is below 15%, which belongs to the skin factor. Low relative error, narrowed uncertainty range over time, and excellent graphical match obtained between the simulated derivative data and the generated curve by using the iterative EnKF verify the robustness of the developed algorithm even in dealing with complex heterogeneous models.